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Machin-like Formula
In mathematics, Machin-like formulae are a popular technique for computing to a large number of digits. They are generalizations of John Machin's formula from 1706: :\frac = 4 \arctan \frac - \arctan \frac which he used to compute to 100 decimal places. Machin-like formulas have the form where c_0 is a positive integer, c_n are signed non-zero integers, and a_n and b_n are positive integers such that a_n < b_n. These formulas are used in conjunction with the expansion for : Derivation The[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arctangent Areas
In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry. Notation Several notations for the inverse trigonometric functions exist. The most common convention is to name inverse trigonometric functions using an arc- prefix: , , , etc. (This convention is used throughout this article.) This notation arises from the following geometric relationships: when measuring in radians, an angle of ''θ'' radians will correspond to an arc whose length is ''rθ'', where ''r'' is the radius of the circle. Thus in the u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Denominator
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A ''common'', ''vulgar'', or ''simple'' fraction (examples: \tfrac and \tfrac) consists of a numerator, displayed above a line (or before a slash like ), and a non-zero denominator, displayed below (or after) that line. Numerators and denominators are also used in fractions that are not ''common'', including compound fractions, complex fractions, and mixed numerals. In positive common fractions, the numerator and denominator are natural numbers. The numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or a whole. The denominator cannot be zero, because zero parts can never make up a whole. For example, in the fraction , the numerator 3 indicates that the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lenstra–Lenstra–Lovász Lattice Basis Reduction Algorithm
The Lenstra–Lenstra–Lovász (LLL) lattice basis reduction algorithm is a polynomial time lattice reduction algorithm invented by Arjen Lenstra, Hendrik Lenstra and László Lovász in 1982. Given a basis \mathbf = \ with ''n''-dimensional integer coordinates, for a lattice L (a discrete subgroup of R''n'') with d \leq n , the LLL algorithm calculates an ''LLL-reduced'' (short, nearly orthogonal) lattice basis in time \mathcal O(d^5n\log^3 B) where B is the largest length of \mathbf_i under the Euclidean norm, that is, B = \max\left(\, \mathbf_1\, _2, \, \mathbf_2\, _2, \dots, \, \mathbf_d\, _2\right). The original applications were to give polynomial-time algorithms for factorizing polynomials with rational coefficients, for finding simultaneous rational approximations to real numbers, and for solving the integer linear programming problem in fixed dimensions. LLL reduction The precise definition of LLL-reduced is as follows: Given a basis \mathbf=\, define its Gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carl Størmer
Fredrik Carl Mülertz Størmer (3 September 1874 – 13 August 1957) was a Norwegian mathematician and astrophysicist. In mathematics, he is known for his work in number theory, including the calculation of and Størmer's theorem on consecutive smooth numbers. In physics, he is known for studying the movement of charged particles in the magnetosphere and the formation of aurorae, and for his book on these subjects, ''From the Depths of Space to the Heart of the Atom''. He worked for many years as a professor of mathematics at the University of Oslo in Norway. A crater on the far side of the moon is named after him. Personal life and career Størmer was born on 3 September 1874 in Skien, the only child of a pharmacist Georg Ludvig Størmer (1842–1930) and Elisabeth Amalie Johanne Henriette Mülertz (1844–1916). His uncle was the entrepreneur and inventor Henrik Christian Fredrik Størmer. Størmer studied mathematics at the Royal Frederick University in Kristiania, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kikuo Takano
was a Japanese poet and mathematician. He was born on Sado Island in 1927. He graduated from Utsunomiya Agricultural College in 1948. He began to write poems from the day after Japan had ended its role in World War II. Being inspired from surrealism and Heidegger, he wrote poems that ask the meaning of being In metaphysics, ontology is the philosophical study of being, as well as related concepts such as existence, becoming, and reality. Ontology addresses questions like how entities are grouped into categories and which of these entities .... He was part of the VOU and Arechi poetry groups, and was awarded the Attilio Bertolucci Award for his poetry. Takano also discovered a Machin-like formula for calculating pi. Selected bibliography * * References {{DEFAULTSORT:Takano, Kikuo 20th-century Japanese mathematicians 21st-century Japanese mathematicians People from Niigata Prefecture 2006 deaths 1927 births 20th-century Japanese poets ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supercomputer
A supercomputer is a computer with a high level of performance as compared to a general-purpose computer. The performance of a supercomputer is commonly measured in floating-point operations per second (FLOPS) instead of million instructions per second (MIPS). Since 2017, there have existed supercomputers which can perform over 1017 FLOPS (a hundred quadrillion FLOPS, 100 petaFLOPS or 100 PFLOPS). For comparison, a desktop computer has performance in the range of hundreds of gigaFLOPS (1011) to tens of teraFLOPS (1013). Since November 2017, all of the world's fastest 500 supercomputers run on Linux-based operating systems. Additional research is being conducted in the United States, the European Union, Taiwan, Japan, and China to build faster, more powerful and technologically superior exascale supercomputers. Supercomputers play an important role in the field of computational science, and are used for a wide range of computationally intensive tasks in v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hitachi, Ltd
() is a Japanese multinational conglomerate corporation headquartered in Chiyoda, Tokyo, Japan. It is the parent company of the Hitachi Group (''Hitachi Gurūpu'') and had formed part of the Nissan ''zaibatsu'' and later DKB Group and Fuyo Group of companies before DKB and Fuji Bank (the core Fuyo Group company) merged into the Mizuho Financial Group. As of 2020, Hitachi conducts business ranging from IT, including AI, the Internet of Things, and big data, to infrastructure. Hitachi is listed on the Tokyo Stock Exchange and Nagoya Stock Exchange and its Tokyo listing is a constituent of the Nikkei 225 and TOPIX Core30 indices. It is ranked 38th in the 2012 Fortune Global 500 and 129th in the 2012 Forbes Global 2000. History Hitachi was founded in 1910 by electrical engineer Namihei Odaira (1874–1951) in Ibaraki Prefecture. The company's first product was Japan's first induction motor, initially developed for use in copper mining. The company began as an in-house ven ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tokyo University
, abbreviated as or UTokyo, is a public research university located in Bunkyō, Tokyo, Japan. Established in 1877, the university was the first Imperial University and is currently a Top Type university of the Top Global University Project by the Japanese government. UTokyo has 10 faculties, 15 graduate schools and enrolls about 30,000 students, about 4,200 of whom are international students. In particular, the number of privately funded international students, who account for more than 80%, has increased 1.75 times in the 10 years since 2010, and the university is focusing on supporting international students. Its five campuses are in Hongō, Komaba, Kashiwa, Shirokane and Nakano. It is considered to be the most selective and prestigious university in Japan. As of 2021, University of Tokyo's alumni, faculty members and researchers include seventeen prime ministers, 18 Nobel Prize laureates, four Pritzker Prize laureates, five astronauts, and a Fields Medalist. His ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yasumasa Kanada
was a Japanese computer scientist most known for his numerous world records over the past three decades for calculating digits of . He set the record 11 of the past 21 times. Kanada was a professor in the Department of Information Science at the University of Tokyo in Tokyo, Japan Japan ( ja, 日本, or , and formally , ''Nihonkoku'') is an island country in East Asia. It is situated in the northwest Pacific Ocean, and is bordered on the west by the Sea of Japan, while extending from the Sea of Okhotsk in the north ... until 2015. From 2002 until 2009, Kanada held the world record calculating the number of digits in the decimal expansion of pi – exactly 1.2411 trillion digits. The calculation took more than 600 hours on 64 nodes of a HITACHI SR8000/MPP supercomputer. Some of his competitors in recent years include Jonathan Borwein, Jonathan and Peter Borwein and the Chudnovsky brothers. See also * Chronology of computation of π, Chronology of computation of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in many other branches of mathematics such as analytic number theory, complex analysis, and infinitesimal calculus. He introduced much of modern mathematical terminology and Mathematical notation, notation, including the notion of a function (mathematics), mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy and music theory. Euler is held to be one of the greatest mathematicians in history and the greatest of the 18th century. A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all." Carl Friedrich Gauss remarked: "The study of Euler's works will remain the best school for the different fields of mathematics, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Approximations Of Pi
An approximation is anything that is intentionally similar but not exactly equal to something else. Etymology and usage The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix ''ad-'' (''ad-'' before ''p'' becomes ap- by assimilation) meaning ''to''. Words like ''approximate'', ''approximately'' and ''approximation'' are used especially in technical or scientific contexts. In everyday English, words such as ''roughly'' or ''around'' are used with a similar meaning. It is often found abbreviated as ''approx.'' The term can be applied to various properties (e.g., value, quantity, image, description) that are nearly, but not exactly correct; similar, but not exactly the same (e.g., the approximate time was 10 o'clock). Although approximation is most often applied to numbers, it is also frequently applied to such things as mathematical functions, shapes, and physical laws. In science, approximation can refer to u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
